A garment worn under the clothes used to "lift and separate" a female's breasts. The names comes from bracière, the old french word referring to arm protector. The brassiere replaced corsets as the female undergarment around the 1900s, when it was most likely developed by Herminie Cadolle. This term is now archaic however and the garment is simply referred to as a "bra," which is french for "arm". Brassiere is only useful when one needs to refer to bras in code:

Eliza: So Laura do you want to go shopping for brassieres tomorrow?
Laura: Alright, that sounds good.
Mike: What's a brassiere?
Eliza: ... really long gloves. It means arm protector in french.